Definably Compact Abelian Groups

Edmundo, Mário J.; Otero, Margarita

doi:10.1142/s0219061304000358

Cited by 58 publications

(96 citation statements)

References 9 publications

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“…For example, from [16] we know that if a definable group is compactly dominated then it has the fsg property and a unique invariant Keisler measure. Of course the proof of Theorem 8.1 (or statement ( * )) depends on G having the fsg property as well as the knowledge of torsion points from [8] (for definably compact G). It would be interesting to try to recover the torsion points statement directly from compact domination.…”

Section: The Final Point Is Easymentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

151

269

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Abstract. We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

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Section: The Final Point Is Easymentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

151

269

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“…The following consequence of Lemma 2.7 is proved in exactly the same way as its definable analogue in [4] Corollary 2.9. Below, for w : W −→ G a locally definable covering map, we say that W is definably w-connected if there is no proper locally definable subset of W which is both w-open and wclosed and whose intersection with any definable subset of W is definable.…”

Section: Preliminary Resultsmentioning

confidence: 83%

“…Thus we generalize the theory of [3] and [4] Section 2 to the category of locally definable covering maps of locally definable groups in R. Since the arguments are similar we will omit the details. Definition 2.1 A set Z is a locally definable set over A, where A ⊆ R and |A| < ℵ 1 , if there is a countable collection {Z i : i ∈ I} of definable subsets of R n , all definable over A, such that:…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…In our category, the path and the homotopy liftings can be proved as in [4] by observing that, by saturation, a definable subset of G is covered by finitely many open definable subsets of G.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…When G is a definably compact, abelian definable group, we conjecture that l above is the dimension of G. This is known to be the case when R is linear ( [5]) or R is an o-minimal expansion of a real closed field ( [4]). So the conjecture is open for R eventually linear but not linear.…”

Section: Definition 12 the Category Cov(g) And Its Full Subcategory Covmentioning

confidence: 99%

See 2 more Smart Citations

The universal covering homomorphism in o‐minimal expansions of groups

Edmundo

Eleftheriou

2007

Mathematical Logic Qtrly

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Suppose that G is a definably connected, definable group in an ominimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p : G −→ G is a locally definable covering homomorphism and π 1 (G) is isomorphic to the o-minimal fundamental group π(G) of G defined using locally definable covering homomorphisms. * With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU).

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A remark on divisibility of definable groups

Edmundo

2005

MLQ - Math. Log. Quart.

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We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an o-minimal expansion of a field, k ∈ N and p k : G −→ G is the definable map given by p k (x) = x k for all x ∈ G, then for all x ∈ G, we have |(p k ) −1 (x)| ≥ k r where r > 0 is the maximal dimension of abelian definable subgroups of G. * With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU). MSC: 03C64; 20E99.

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Definably Compact Abelian Groups

Cited by 58 publications

References 9 publications

On NIP and invariant measures

On NIP and invariant measures

The universal covering homomorphism in o‐minimal expansions of groups

A remark on divisibility of definable groups

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